1. 11. Market microstructure: information-based models
11.1 Introduction
The idea: price is a source of information that investors can use for their trading decisions. For example, if security price falls, investors may suggest that price will further deteriorate and refrain from buying this security (which contradicts the Walrasian framework).
Rational expectations: informed traders and market makers behave rationally in the sense that all their actions are focused on maximizing their wealth (or some utility function in the case of risk-averse agents). Market in these models reaches an equilibrium state that satisfies participants’ expectations.
Informed investors trade only on one side of the market at any given time. Hence market makers face adverse selection problem.

2. 11. Market microstructure: information-based models
11.2 Kyle’s model for batch auction (1985)
One-period model
- single risky security with value v = N(p0, σ02)
- risk-neutral informed trader (insider) submits strategic order of size x(v)
- several liquidity traders submit orders randomly with size y = N(0, σy2)
- risk-neutral market maker (dealer) faces demand z = x + y and
assumes that clearing price is linear upon demand: p = λz + μ
λ is inverse liquidity.
Expected insider’s profit equals: E[x(p – v)] = x(v – λx - μ).
Then optimal insider’s demand equals x = (v – μ)/2λ.
In equilibrium, μ* = p0 and λ* = σ0/2σy

3. 11. Market microstructure: information-based models
11.2 Kyle’s model (continued)
Order flow from liquidity traders distorts the insider’s information: even if insider knows that the traded security is overpriced and hence is not interested in buying it, a large idiosyncratic buy order from a liquidity trader will motivate the dealer to increase price.
Insider’s profit = loss of liquidity traders = 0.5σyσ0
Dealer’s profit = 0 (Bertrand’s competition model)
Why would anyone run batch auction?
Multi-period model: K auctions at times tk = kΔt = k/K
Liquidity trading volume: Δyk = N(0, σy2Δt)
Insider’s demand: Δxk = βk(v - pk) Δt
Dealer’s price: pk = pk-1 + λk(Δxk + Δyk)
No analytical solution…

4. 11. Market microstructure: information-based models
11.3 Glosten-Milgrom model for a continous market (1985)
(see Hasbrouck (2007))
Sequential trading model: risk-neutral dealer trades with insiders arriving with probability μ and liquidity traders with probability 1 - μ, one unit at a time. Security can have two values: high vH (good news) with probability 1-θ and low vL (bad news) with probability θ. Insiders trade according to news but liquidity traders buy and sell with probability of 0.5.

5. 11. Market microstructure: information-based models
11.3 Glosten-Milgrom model (continued)
Dealer sets “regret-free” prices, i.e. these prices are dealer’s expectations of the security’s value based on trading signals (B=buy, S=sell)
ask price a = E[V | B] = VL Pr(V= VL | B) + VH Pr(V=VH | B)
bid price b = E[V | S] = VL Pr(V= VL | S) + VH Pr(V=VH | S)
Bayes’ rule: Pr(A|B) =
Pr(B) = Pr(V= VL)Pr(B |V= VL) + Pr(V= VH)Pr(B |V= VH) = 0.5(1 + μ(1-2θ))
Pr(V= VL|B) =

6. 11. Market microstructure: information-based models
11.3 Glosten-Milgrom model (continued 2)
ask =
bid =
spread = => μ(VH -VL) when θ = 0.5
Spread grows with the number of insiders: adverse selection

7. 11. Market microstructure: information-based models
11.4 Extensions of Glosten-Milgrom model
Easley & O’Hara (1987; 1992); Subrahmanyam (1991); Back & Baruch (2004)
- “No-news” case: only liquidity traders trade => decreases spread.
- Orders of small and large sizes. Large orders yield higher gains to
informed traders but reveal their private information to the market.
On the other hand, small orders help to hide some private information and
improve price for large orders. Hence informed traders face a choice
between trading exclusively large orders and mixing them with small
ones.
- Multiple insiders. With their growing number, liquidity first decreases due to
risk aversion. However, with further number growth, liquidity starts increasing.

8. 11. Market microstructure: information-based models
11.4 Summary
Price is a source of information that investors can use for their trading decisions. For example, if security price falls, investors may suggest that price will further deteriorate and refrain from buying this security.
Market makers face adverse selection problem while trading with informed investors who trade only on one side of the market at any given time.
Dealers and informed investors behave rationally, i.e. make trading decisions to maximize their wealth (or utility function depending on wealth in case risk-averse agents). As a result, price converges to some equilibrium value that satisfies rational expectations.
In the batch auctions, dealers compensate potential losses from adverse selection by setting asset price increasing with demand (Kyle’s model (1985).
In the sequential trade models, dealers compensate potential losses from adverse selection by setting the bid/ask spread that grows with the number of informed traders and with order size (Glosten-Milgrom model (1985)).